Properties

Label 3332.1.o.f
Level $3332$
Weight $1$
Character orbit 3332.o
Analytic conductor $1.663$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -68
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 476)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.188737808.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{2} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{3} + \zeta_{12}^{4} q^{4} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{6} - q^{8} + ( -1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{9} +O(q^{10})\) \( q + \zeta_{12}^{2} q^{2} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{3} + \zeta_{12}^{4} q^{4} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{6} - q^{8} + ( -1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{9} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{11} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{12} - q^{13} -\zeta_{12}^{2} q^{16} -\zeta_{12}^{4} q^{17} + ( 1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{18} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{22} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{24} + \zeta_{12}^{4} q^{25} -\zeta_{12}^{2} q^{26} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{27} -\zeta_{12}^{4} q^{32} + ( -1 - 2 \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{33} + q^{34} + ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{36} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{39} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{44} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{48} - q^{50} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{51} -\zeta_{12}^{4} q^{52} -\zeta_{12}^{4} q^{53} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{54} + q^{64} + ( 1 - \zeta_{12}^{2} - 2 \zeta_{12}^{4} ) q^{66} + \zeta_{12}^{2} q^{68} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{71} + ( 1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{72} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{75} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{78} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{79} + \zeta_{12}^{4} q^{81} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{88} + \zeta_{12}^{2} q^{89} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{96} + ( -2 \zeta_{12} + 2 \zeta_{12}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 4 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 4 q^{9} - 4 q^{13} - 2 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{25} - 2 q^{26} + 2 q^{32} - 6 q^{33} + 4 q^{34} + 8 q^{36} - 4 q^{50} + 2 q^{52} + 2 q^{53} + 4 q^{64} + 6 q^{66} + 2 q^{68} + 4 q^{72} - 2 q^{81} + 2 q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.500000 + 0.866025i −0.866025 + 1.50000i −0.500000 + 0.866025i 0 −1.73205 0 −1.00000 −1.00000 1.73205i 0
67.2 0.500000 + 0.866025i 0.866025 1.50000i −0.500000 + 0.866025i 0 1.73205 0 −1.00000 −1.00000 1.73205i 0
2039.1 0.500000 0.866025i −0.866025 1.50000i −0.500000 0.866025i 0 −1.73205 0 −1.00000 −1.00000 + 1.73205i 0
2039.2 0.500000 0.866025i 0.866025 + 1.50000i −0.500000 0.866025i 0 1.73205 0 −1.00000 −1.00000 + 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
4.b odd 2 1 inner
7.c even 3 1 inner
17.b even 2 1 inner
28.g odd 6 1 inner
119.j even 6 1 inner
476.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.o.f 4
4.b odd 2 1 inner 3332.1.o.f 4
7.b odd 2 1 476.1.o.c 4
7.c even 3 1 3332.1.g.f 2
7.c even 3 1 inner 3332.1.o.f 4
7.d odd 6 1 476.1.o.c 4
7.d odd 6 1 3332.1.g.g 2
17.b even 2 1 inner 3332.1.o.f 4
28.d even 2 1 476.1.o.c 4
28.f even 6 1 476.1.o.c 4
28.f even 6 1 3332.1.g.g 2
28.g odd 6 1 3332.1.g.f 2
28.g odd 6 1 inner 3332.1.o.f 4
68.d odd 2 1 CM 3332.1.o.f 4
119.d odd 2 1 476.1.o.c 4
119.h odd 6 1 476.1.o.c 4
119.h odd 6 1 3332.1.g.g 2
119.j even 6 1 3332.1.g.f 2
119.j even 6 1 inner 3332.1.o.f 4
476.e even 2 1 476.1.o.c 4
476.o odd 6 1 3332.1.g.f 2
476.o odd 6 1 inner 3332.1.o.f 4
476.q even 6 1 476.1.o.c 4
476.q even 6 1 3332.1.g.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.1.o.c 4 7.b odd 2 1
476.1.o.c 4 7.d odd 6 1
476.1.o.c 4 28.d even 2 1
476.1.o.c 4 28.f even 6 1
476.1.o.c 4 119.d odd 2 1
476.1.o.c 4 119.h odd 6 1
476.1.o.c 4 476.e even 2 1
476.1.o.c 4 476.q even 6 1
3332.1.g.f 2 7.c even 3 1
3332.1.g.f 2 28.g odd 6 1
3332.1.g.f 2 119.j even 6 1
3332.1.g.f 2 476.o odd 6 1
3332.1.g.g 2 7.d odd 6 1
3332.1.g.g 2 28.f even 6 1
3332.1.g.g 2 119.h odd 6 1
3332.1.g.g 2 476.q even 6 1
3332.1.o.f 4 1.a even 1 1 trivial
3332.1.o.f 4 4.b odd 2 1 inner
3332.1.o.f 4 7.c even 3 1 inner
3332.1.o.f 4 17.b even 2 1 inner
3332.1.o.f 4 28.g odd 6 1 inner
3332.1.o.f 4 68.d odd 2 1 CM
3332.1.o.f 4 119.j even 6 1 inner
3332.1.o.f 4 476.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3}^{4} + 3 T_{3}^{2} + 9 \)
\( T_{5} \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 9 + 3 T^{2} + T^{4} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( ( 1 - T + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 1 - T + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( ( -3 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( 9 + 3 T^{2} + T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 1 - T + T^{2} )^{2} \)
$97$ \( T^{4} \)
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